As the Principia is pretty ugly and tortured at least for me let me offer:

Naive set theory. Halmos, Paul R. http://people.whitman.edu/~guichard/260/halmos__naive_set_th...

Note the first entry of "Ordered Pairs"

> What does it mean to arrange the elements of a set A in some order?

Also note how the earlier section on "Unordered Pairs" is more about building the axiom of pairing etc...to get to ordered pairs which gets to the Cartesian product, which outputs ordered pairs.

It doesn't matter if you go through Zermelo's theorem+Zorn, that states that every set can be well-ordered, or though Cartesian product's and/or AC. (Note: This is in FoL well-ordering and AC are the same, but not in SoL and HoL)

It is not that sets are expressly unordered, as a set of points in a line segment would very much have an order, but that you didn't actively arrange the elements in order to take advantage of properties that are useful to you.

Maybe I just hit mental blocks but IMHO it is important that when you make the assumption that "there exists a set." it is very important to realize that it is "unordered" because you haven't imposed one, but is not an innate property of an element of the set.

Hopefully that helps in addressing this from your original post.

> "ordered pair" is not part of set theory

While many creators of both naive and formal set theories may choose to not define (a,b) = {{a},{a,b}} explicitly, the output of the Cartesian product is the ordered pairs, so it doesn't matter, you don't have a useful set theory without them.